In this paper, a predator-prey system with Holling type function response incorporating prey refuge is presented.
The random use model, the fixed total range model, incorporated all refuge sites within a circle of radius h.
Chen and Chen [10] proposed a ratio-dependent predator-prey model incorporating a prey refuge.
In this paper, we have considered a predator-prey system with a general functional response incorporating a prey refuge.
A predator-prey model with simplified Holling type III response function incorporating a prey refuge under sparse effect is considered.
The first system is related to the photosynthetic production of phytoplankton, the second with a time delayed prey predator system incorporating a prey refuge.
During the last two decades, the study of dynamic behaviors of predator-prey system incorporating a prey refuge become one of the most important research topic, see [1 45] and the references cited therein.
Huang et al. [9] studied the stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge: d x d t = a x - b x 2 - α x 2 y β 2 + x 2, d y d t = - c y + k α x 2 y β 2 + x 2. (1.1).
Motivated by the study of Huang et al. [9] and Ji and Wu [10], we consider the following predator-prey model with Holling type III response function incorporating a prey refuge under sparse effect: d x d t = r x 2 ( 1 - x k ) - ( 1 - m ) 2 x 2 y a + ( 1 - m ) 2 x 2, d y d t = y [ - d + u ( 1 - m ) 2 x 2 a + ( 1 - m ) 2 x 2 ]. (1.2).
In [26], Tripathi et al. studied a predator-prey model with Beddington-DeAngelis type functional response incorporating a prey refuge, that is, textstylebegin{cases} frac{du}{dt}= u 1-u-frac{a 1-m-m) v}{1-u-frac{a 1-m} 1-u-frac{a 1-m}=v ( frac{e(1-u-frac{a 1-mm) u+c v}{theta), end{cases} (1+b 1-mth the initial conditions (u(0)=u_{0}>0), (v(0)=v_{0}>0), which are biologically meaningful.
